Triadic judgment models and Weber's law

Let a, b, c, with a?b?c, be positive real numbers indicating the intensities of physical stimuli in a psychophysical experiment; let P_abc be the probability that b is judged to be more similar to a (“closer to”) a than to c. This paper investigates the following representation and its subcases for triadic judgments

P_abc=F[u(a)-u(b),u(b)-u(c)],     

Balance,positivity, and agreement in the Jordan Paradigm: A defense of balance theory

It was argued that Heider's p?o?x triad can best be conceptualized according to a three- factor analysis of variance model in which the p?o, p?x, and o?x bands are all factors. From this perspective the balance effect is the triple interaction, the positivity effect a main effect for the p?o factor, and the agreement effect a p?x by o?x interaction. Although the existence of the latter two effects has previously been regarded as damaging to balance theory, it was shown that these effects could be interpreted from a balance perspective and that balance theory could be used to generate supportable propositions regarding these effects. Thus in agreement with a unit-relation interpretation it was shown, in accordance with balance theory, that positivity effects are obtained when the subject, or p, assumes future contact with o, that reverse positivity effects are obtained when the subject anticipates breaking off contact with o, and that no positivity effect is obtained when there is absolutely no contact, past or future. It was also demonstrated in an experiment involving the p?o?q triad that, in accordance with balance theory, positivity effects may be produced by the assumption or inference of same-sign reciprocation in sentiment. Evidence for two balance processes underlying agreement effects was also found. One of these processes was based on the assumption that the subject would have or reveal psycho-logical reasons for the disagreement and thus produce imbalance. Consistent with this interpretation it was found that the agreement effect was significantly larger when future contact with discussion of x was assumed than when future contact without discussion of x was assumed. The other, or unit-relation interpretation was supported by evidence indicating that the breaking off of contact resulted in a reversed agreement effect. In general, it was argued that balance theory did quite well in such phenomenological investigations when attention was not narrowly restricted to the three-sign pattern but considered other potential cognitive bands.     

Comparing the meanings of “if” and “all”

In this study, we compared the everyday meanings of conditionals (“if p then q”) and universally quantified statements (“all p are q”) when applied to sets of elements. The interpretation of conditionals was predicted to be directly related to the conditional probability, such that P(“if p then q”) = P(q|p). Quantified statements were assumed to have two interpretations. According to an instance-focused interpretation, quantified statements are equivalent to conditionals, such that P(“all p are q”) = P(q|p). According to a set-focused interpretation, “all p are q” is true if and only if every instance in set p is an instance of q, so that the statement would be accepted when P(q|p) = 1 and rejected when this probability was below 1. We predicted an instance-focused interpretation of “all” when the relation between p and q expressed a general law for an infinite set of elements. A set-focused interpretation of “all” was predicted when the relation between p and q expressed a coincidence among the elements of a finite set. Participants were given short context stories providing information about the frequency of co-occurrence of cases of p, q, not-p, and not-q in a population. They were then asked to estimate the probability that a statement (conditional or quantified) would be true for a random sample taken from that population. The probability estimates for conditionals were in accordance with an instance-focused interpretation, whereas the estimates for quantified statements showed features of a set-focused interpretation. The type of the relation between p and q had no effect on this outcome.     

The Limits of Consent

This journal has frequently taken the position that consent, or at least informed consent, is all that from a secular viewpoint is necessary for an activity to be ethical. We argue to the contrary, that consent is and only is a political criterion for determining criminality—even for a libertarian. Consensual behavior can be unethical—although it should not be criminalized—if the consent will never be truly revocable in the future or if such revocability is severely compromised. We give three examples, one from common experience, and two from the areas normally covered in this journal.     

Generalized procrustes analysis

SupposeP _i ⁽ⁱ⁾ (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σ _i=1 ^m Σ _i=1 ⁿ Δ²(P _j ⁽ⁱ⁾ G _i ), whereG _i is the centroid of them pointsP _i ⁽ⁱ⁾ (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.     

What are numbers?

This paper argues that (cardinal) numbers are originally given to us in the context ‘Fs exist n-wise’, and accordingly, numbers are certain manners or modes of existence, by addressing two objections both of which are due to Frege. First, the so-called Caesar objection will be answered by explaining exactly what kind of manner or mode numbers are. And then what we shall call the Functionality of Cardinality objection will be answered by establishing the fact that for any numbers m and n, if there are exactly m Fs and also there are exactly n Fs, then m = n.     

Confucian Ritual as Body Language of Self, Society, and Spirit

This article explains how li ?Y or ??ritual propriety?? is the ??body language?? of ren ?? or the authentic expression of our humanity. Li and ren are interdependent aspects of a larger creative human way (rendao ???) that can be conceptually distinguished as follows: li refers to the ritualized social form of appropriate conduct and ren to the more general, authentically human spirit this expresses. Li is the social instrument for self-cultivation and the vehicle of harmonious human interaction. More, li must mean something that is effectively communicated to others for an authentic, human (ren) interaction to occur. Li is the body language of ren in being the ritual vehicle for its?? expression; however, li is underdetermined by ren and so must be distinguished from it in on further grounds: authentic human activity must not just be equivocated with social convention because conclusively establishing whether a particular action is li (or is a truly ren action) is impossible. As a result, li is often confused with social power and privilege that is easier to empirically identify than ren conduct is, but this is a mistake since li has to express ren or it is not li at all. The inescapable ambiguity of li ?C an ambiguity that attaches to any language ?C can be critiqued by the Western view that sees something ??essential?? to the ??self,?? and that makes one a ??self?? in and of oneself and not in a way that depends on others. I show that such Western individualism ?C while resting on a fundamentally different way of thinking of being a person and living a good life ?C does not reduce Confucian ritual to being an instrument for social discrimination and subordination. My argument is indebted to twentieth-century philosophy of language in the West that offered the idea that some words are actions.     

Compatible Operations on Residuated Lattices

This work extend to residuated lattices the results of [7]. It also provides a possible generalization to this context of frontal operators in the sense of [9]. Let L be a residuated lattice, and f : L ^k ?? L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L. We use this characterization of compatible functions in order to prove that the variety of residuated lattices is locally affine complete. We study some compatible functions on residuated lattices which are a generalization of frontal operators. We also give conditions for two operations P(x, y) and Q(x, y) on a residuated lattice L which imply that the function ${x \mapsto min\{y \in L : P(x, y) \leq Q(x, y)\}}$ when defined, is equational and compatible. Finally we discuss the affine completeness of residuated lattices equipped with some additional operators.     

Cut-elimination for Weak Grzegorczyk Logic Go

We present a syntactic proof of cut-elimination for weak Grzegorczyk logic Go. The logic has a syntactically similar axiomatisation to Gödel–Löb logic GL (provability logic) and Grzegorczyk’s logic Grz. Semantically, GL can be viewed as the irreflexive counterpart of Go, and Grz can be viewed as the reflexive counterpart of Go. Although proofs of syntactic cut-elimination for GL and Grz have appeared in the literature, this is the first proof of syntactic cut-elimination for Go. The proof is technically interesting, requiring a deeper analysis of the derivation structures than the proofs for GL and Grz. New transformations generalising the transformations for GL and Grz are developed here.     

Strategies used by children when solving simple subtractions

Ten children, 9–11 years old, solved all subtraction problems in the form of M?N=…, where 0 ? M ? 13, 0 ? N ? 13 and M ? N. The solution times were analysed and used for the formulation of a process model for subtraction. The model involves memory processes on two different levels, called reproductive and reconstructive respectively. When M=N, N=1, and M=2N the answers were quickly retrieved in reproductive memory processes. The reconstructive processes were found to be analogous to one of two counting procedures, viz. counting up and counting down. In general, the counting process starts either on N (when M < 2N) and counts up to reach the answer, or on M (when M > 2N) and counts down to reach the answer. This may reflect an effort to minimize the number of steps to be counted. However, when M > 10 and N < 10 a problem is always solved in a decrementing counting process. When M=10 many subtractions are solved in a reproductive memory process and the number 10 is also important as a point of reference for solving subtractions when M > 10 and N < 10.     

A generalization of Piéron's law to include background intensity and latency distribution

Piéron's law, which formulates the effect of the signal intensity i on the mean or the median reaction time (MRT) as a power function, MRT=R+i^-βC, in which R, β and C are positive parameters, is of much interest in the simple reaction time literature. In particular, consistent estimates of β are obtained. However, this law offers no information regarding the effect of the signal intensity on other aspects of the reaction time distribution. In addition, the background intensity, which also has some impact on the detectability of the signal, is not considered in Piéron's law. In this paper, we generalize Piéron's law based on the above comments. That is, we assume that both β and C are functions of the background intensity and of the percentile rank. We then investigate from a theoretical viewpoint some possible functional forms of β and C, with a special emphasis on the dependency of β on the background intensity. It is shown that the so-called “Fechner-Thurstone-type representation” forces β and C to be “separable.” If we further assume a near-miss to Weber's law-type representation, then the functional form of β can be determined.     

An approximation of theK outN reliability of a test,and a scoring procedure for determining which items an examinee knows

Consider any scoring procedure for determining whether an examinee knows the answer to a test item. Letx _i = 1 if a correct decision is made about whether the examinee knows the ith item; otherwisex _i = 0. Thek out ofn reliability of a test isρ _k = Pr (Σx _i ≥k). That is,ρ _k is the probability of making at leastk correct decisions for a typical (randomly sampled) examinee. This paper proposes an approximation ofρ _k that can be estimated with an answer-until-correct test. The paper also suggests a scoring procedure that might be used whenρ _k is judged to be too small under a conventional scoring rule where it is decided an examinee knows if and only if the correct response is given.     

Finite automata and S−R models

Suppes (1969, Journal of Mathematical Psychology, 6) apparently proved that any finite automaton could be mimicked by the asymptotic behavior of a stimulussampling S?R model, broadly implying that cognitive theory could be reduced to S?R theory. Here it is shown that the finite automata used by Suppes are more restricted and less powerful than finite automata in general. Furthermore, the S?R models proposed by Suppes are limited to producing the behavior of only these restricted automata, and cannot mimic all finite automata. Hence, the formal claim that S?R models are adequate to produce all behaviors obtainable from finite automata, and the informal claim that cognitive theory reduces to S?R theory, do not follow from Suppes's (1969) result. Some alternative S?R models and their problems are also briefly discussed.     

The evolutionary restriction of aggression within a species: A game theory analysis

Can selection based on individual advantage result in limits to the magnitude of the damage animals may inflict on conspecifics in agonistic encounters? A formulation of intraspecific conflict in terms of game theory introduced by Maynard Smith (Journal of Theoretical Biology, 1974, 47, 209–221) is used to examine this question. Maynard Smith put forward the concept of an evolutionarily stable strategy (ESS), one which it will never be to an individual's advantage to depart from, and the resulting analysis is used here to determine the behaviors that will evolve when aggression is modeled as a game between two contestants, each of which has two options, “Attack” or “Display.” It is assumed that the opponents may be of two types which differ in the destruction they are capable of inflicting but that prior to a contest a contestant cannot recognize the type its opponent belongs to. The Weak inflict damage D_I when victorious in a contest but the Strong inflict damage D_K > D_I and the Strong always defeat the Weak when both attack. It is shown that under selection the Strong will wholly replace the Weak when D_K ≤ V, V being the value of the prize, but that when D_K > V an equilibrium between the two types results. Such a heterogeneous group gives a greater return to its members than a homogeneous group with D ≤ V, and the value of D_K will tend to become large, the probability of an attack decreasing inversely. A population in which two types are initially present but recognizably different will evolve toward indistinguishability. If a third type is introduced the intermediate strain will disappear, restoring a bimodal distribution of destructive capacity. In a group whose members are related there will be a higher proportion of the Weak type at equilibrium and the overall probability of attack will be reduced. Such a group will offer a greater return to its members than association with randomly selected conspecifics, assuming that the ESS is played in each case. Under such conditions evolution should favor association with kin. In all cases V constitutes an upper limit to the expected level of damage in encounters. This defines an intrinsic restriction to the magnitude or the frequency of aggressive damage.     

The Nonsingularity of Γ in Covariance Structure Analysis of Nonnormal Data

Covariance structure analysis of nonnormal data is important because in practice all data are nonnormal. When applying covariance structure analysis to nonnormal data, it is generally assumed that the asymptotic covariance matrix Γ for the nonredundant terms in the sample covariance matrix S is nonsingular. It is shown this need not be the case, which raises a question of how restrictive this assumption may be and how difficult it may be to verify it. It is shown that Γ is nonsingular whenever sampling is from a nonsingular distribution, including any distribution defined by a density function. In the discrete case necessary and sufficient conditions are given for the nonsingularity of Γ, and it is shown how to demonstrate Γ is nonsingular with high probability. Thus, the nonsingularity of Γ assumption is mild and one should feel comfortable about making it. These observations also apply to the asymptotic covariance matrix Γ that arises in structural equation modeling.     

The Bayesian explanation of transmission failure

Even if our justified beliefs are closed under known entailment, there may still be instances of transmission failure. Transmission failure occurs when P entails Q, but a subject cannot acquire a justified belief that Q by deducing it from P. Paradigm cases of transmission failure involve inferences from mundane beliefs (e.g., that the wall in front of you is red) to the denials of skeptical hypotheses relative to those beliefs (e.g., that the wall in front of you is not white and lit by red lights). According to the Bayesian explanation, transmission failure occurs when (i) the subject’s belief that P is based on E, and (ii) P(Q|E) < P(Q). But there are compelling cases of transmission failure where P(Q|E) > P(Q). No modifications of the Bayesian explanation are capable of accommodating such cases, so the explanation must be rejected as inadequate. Alternative explanations employing simple subjunctive conditionals are fully capable of capturing all of the paradigm cases, as well as those missed by the Bayesian explanation.     

Developmental changes in conditional reasoning: Linguistic or logical?

This study investigated reasoning with abstract conditional sentences as a function of age. Subjects from third-grade to eleventh-grade were required to evaluate the conclusion of several conditional arguments. The results confirmed the previously established finding that performance improves with age, particularly between 11 and 15 years. This finding could be interpreted to indicate that individuals become more logical as they get older. However, another possible interpretation is that the meaning of a conditional sentence like pthenq for naive Ss may not always be given by the truth function pq true, $\text{p}\text{q}$ false, $\text{p}\text{q}$ true, and $\text{p}\text{q}$ true. Further analysis suggested that at 9 years individuals treat the connective in the sentence, ifpthenq, as if it were either a conjunctive or a biconditional, that the conjunctive meaning disappears with increasing age and after 13 years is gradually superceded by the conditional.     

Cohen’s d needs to be readily interpretable: Comment on Shieh (2013)

Shieh (2013) discussed in detail δ*, a proposed standardized effect size measure for the two-independent-groups design with heteroscedasticity. Shieh focused on inference—notably, the large challenge of calculating confidence intervals for δ*. I contend, however, that the standardizer chosen for δ*, meaning the units in which it is expressed, is appropriate for inference but causes δ* to be inconsistent with conventional Cohen’s d. In addition, δ* depends on the relative sample sizes in the particular experiment and, thus, lacks the generality that is highly desirable if a standardized effect size is to be readily interpretable and also usable in meta-analysis. In the case of heteroscedasticity, I suggest that researchers should choose as standardizer for Cohen’s δ the best available estimate of the SD of an appropriate population, usually the control population, in preference to δ* as discussed by Shieh.     

Doing well enough: Toward a logic for common-sense morality

On the traditional deontic framework, what is required (what morality demands) and what is optimal (what morality recommends) can't be distinguished and hence they can't both be represented. Although the morally optional can be represented, the supererogatory (exceeding morality's demands), one of its proper subclasses, cannot be. The morally indifferent, another proper subclass of the optional-one obviously disjoint from the supererogatory-is also not representable. Ditto for the permissibly suboptimal and the morally significant. Finally, the minimum that morality allows finds no place in the traditional scheme. With a focus on the question, “What would constitute a hospitable logical neighborhood for the concept of supererogation?”, I present and motivate an enriched logical and semantic framework for representing all these concepts of common sense morality.